Result 330, Functional analysis

A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A space can allow arbitrarily accurate finite-dimensional approximations without allowing their amplification of inputs to stay uniformly controlled. This manuscript claims that even a countable collection of uniformly separated points can produce that distinction.

What changes?

The unreviewed manuscript reports a countable uniformly discrete metric space: a set of points with distances between distinct points bounded below by one positive constant. Its real Lipschitz-free space, a Banach space encoding those distances in a linear setting, has the approximation property but not the bounded approximation property. Thus, linear operators with finite-dimensional ranges can approximate the identity uniformly on each compact set, but no such approximation scheme can keep all operator norms below a single finite bound.

What does that help mathematicians do?

The counterexample answers Kalton's question negatively by separating two forms of approximation within this particular class of Banach spaces. A researcher cannot infer bounded approximation merely from the approximation property, even when the underlying metric space is countable and uniformly discrete. Any general criterion guaranteeing bounded approximation here must use additional information that excludes this example, rather than relying on those assumptions alone.

Are there practical applications?

Its immediate value is foundational: it identifies a limit on controlling finite-dimensional approximations in spaces built from metric data. Operator norms measure how much an approximation can magnify an input, so the distinction concerns control as well as accuracy. The reported result is a structural counterexample, not a practical approximation algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

September 26, 2026 21 pages

We construct a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but fails the bounded approximation property, answering Kalton's question negatively. The identity can be approximated on every compact set by finite-rank operators, but no uniform bound on their norms is possible.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026,
  author = {{OpenAI}},
  title = {{A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf}{OAI:Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/330.md.

A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Kalton's question asks whether uniform discreteness of a metric space forces its Lipschitz-free space to have the bounded approximation property. The formalization constructs a countable metric space with all distinct points at distance at least one whose real Lipschitz-free space has the approximation property but fails every bounded approximation bound. The metric space is unbounded and not proper.

The earlier quantitative renorming result for real ℓ1\ell_1 is retained: for each integer p≥1p\ge1, an equivalent complete norm has the 2Bp2B_p-bounded approximation property but fails the pp-bounded approximation property, where Bp=300p2+150p+5B_p=300p^2+150p+5.

Comparator links

Result Comparator statement
Quantitative renorming of real ℓ1\ell_1 RealL1Renorming.lean
Uniformly discrete Lipschitz-free space with AP but no BAP DiscreteLipschitzFree.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.